The Compound Interest Formula: Derivation & Proof
From first-principles arithmetic recurrence relations to continuous exponential limits and finite geometric series.
1. Discrete Compounding Formula
Variable Breakdown & Definitions
| Symbol | Parameter | Engineering Definition | Example |
|---|---|---|---|
| A(t) | Final Accumulated Value | Total capital at horizon t | $100,626 (USD ($)) |
| P | Principal Capital | Starting initial balance | $10,000 (USD ($)) |
| r | Nominal Annual Rate | Annual interest rate expressed in decimal form | 0.08 (Rate (decimal)) |
| n | Compounding Frequency | Number of compounding sub-intervals per year | 12 (Periods / yr) |
| t | Time | Duration in years | 30 (Years) |
*Note: Derived by setting Balance_k = Balance_{k-1} * (1 + r/n). Expanding iteratively yields Balance_k = P * (1 + r/n)^k, where k = n*t.
2. The Continuous Compounding Limit (Euler's e)
As the compounding frequency n approaches infinity (continuous compounding), we evaluate the limit:
lim_(n->∞) P * (1 + r/n)^(n * t) = P * [ lim_(n->∞) (1 + r/n)^(n/r) ]^(r * t) = P * e^(r * t)Where e ≈ 2.718281828... represents Euler's constant. Continuous compounding provides the theoretical upper bound of returns for a given nominal rate.
3. Incorporating Regular Periodic Payments (Annuity)
When regular monthly or annual cash additions (PMT) occur, the total future value is the sum of the principal compounding plus the sum of a finite geometric series for all recurring contributions:
FV = P(1 + i)^N + PMT * [ ((1 + i)^N - 1) / i ]where i = r / n (periodic rate) and N = n * t (total periods).
4. Worked Canonical Example
Evaluating the closed-form recurrence with initial principal $10,000, monthly addition $1,000, nominal annual return 8% (r = 0.08, n = 12), across a 20-year horizon (t = 20):
Illustrative projection. Assumes a constant annual return, contributions at the end of each month, and no taxes, fees, or inflation. Actual returns vary and can be negative. Educational content, not financial advice.
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